On the Strong Solutions of the Nonlinear Viscous Rotating Stratified Fluid

Authors: A. Giniatoulline

Abstract:

A nonlinear model of the mathematical fluid dynamics which describes the motion of an incompressible viscous rotating fluid in a homogeneous gravitational field is considered. The model is a generalization of the known Navier-Stokes system with the addition of the Coriolis parameter and the equations for changeable density. An explicit algorithm for the solution is constructed, and the proof of the existence and uniqueness theorems for the strong solution of the nonlinear problem is given. For the linear case, the localization and the structure of the spectrum of inner waves are also investigated.

Keywords: Galerkin method, Navier-Stokes equations, nonlinear partial differential equations, Sobolev spaces, stratified fluid.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1126543

References:

[1] B. Cushman-Roisin, and J. Beckers, Introduction to Geophysical Fluid Dynamics, New York: Acad. Press, 2011, ch3.
[2] D. Tritton, Physical Fluid Dynamics, Oxford: Oxford UP, 1990, ch.2.
[3] A. Aloyan, “Numerical modeling of remote transport of admixtures in atmosphere,” Numerical Methods in the Problems of Atmospheric Physics and Environment Protection, Novosibirsk: Ac. Sci. USSR, 1985, pp. 55-72.
[4] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, New York: AMS Chelsea Publishing, 2000.
[5] L. Tartar, An Introduction to Navier-Stokes Equations and Oceanography, Berlin: Springer, 2006.
[6] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Zurich: Birkhäuser, 2012.
[7] A. Giniatoulline, and T. Castro, “On the Spectrum of Operators of Inner Waves in a Viscous Compressible Stratified Fluid,” Journal Math. Sci. Univ. of Tokyo, 2012, no. 19, pp. 313-323.
[8] A. Giniatoulline, and T. Castro, “On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows,” Journal of Applied Mathematics and Physics, 2014, no. 2, pp. 528-539.
[9] O. Ladyzhenskaya, The Mathematical Theory of the Viscous Incompressible Flow, New York: Gordon and Breach, 1969.
[10] L. Cattabriga, “Su un Problema al Contorno Relativo al Sistema di Equazioni di Stokes,” Rendiconti del Seminario Matematico della Universita di Padova, 1961, vol. 31, pp. 308-340.
[11] V. Maslennikova, and M. Bogovski, “Elliptic Boundary Value Problems in Unbounded Domains with Noncompact and Nonsmooth Boundaries,” Milan Journal of Mathematics, 1986, no. 56, vol. 1, pp.125-138.
[12] T. Kato, Perturbation theory for Linear Operators, Berlin: Springer, 1966.
[13] S. Agmon, A. Douglis, and L. Nirenberg, “Estimates Near the Boundary for Solutions of Elliptic Differential,” Comm. Pure and Appl. Mathematics, 1964, vol. 17, pp. 35-92.
[14] A. Giniatoulline, “Mathematical Study of Some Models of the Atmosphere Dynamics Counting with Heat Transfer and Humidity,” Recent Advances on Computational Science and Applications, Seoul: WSEAS Press, 2015, vol. 52, pp. 55-61.